Much of the prior mathematical literature has largely focused on the “forward" behavior of branching processes, including the long-run or limiting behavior of such processes [1]}, [2]}. The derivations presented in this article instead help us quantify transient behavior with an efficient and general numerical recipe, extending these methods beyond the special cases for which solutions have been previously established. In particular, finite-time quantities often translate directly to answer relevant clinical questions, and are necessary within inferential schemes [3]}, [4]} such as evaluating the marginal likelihood of data available at a sequence of observation times. Indeed, transition probabilities fully characterize the probabilistic behavior of these stochastic models. Our approach targets these and related quantities applicable to multi-type branching processes beyond our model of mutation here. This contrasts with previous work that for instance posits a neutral or symmetric branching process with specific relationships between the birth and death parameters [5]}; it is worth noting that the more general version of that model was deemed mathematically intractable, analyzed only in supplementary simulations. Our contributions fill a methodological gap suggested in this and related prior literature, which largely resorts to brute-force simulation and thus requires considerable computation time [5]}.
